PATTERN SELECTION IN AN EPIDEMIC MODEL WITH SELF AND CROSS DIFFUSION

2011 ◽  
Vol 19 (01) ◽  
pp. 19-31 ◽  
Author(s):  
WEIMING WANG ◽  
YEZHI LIN ◽  
HAILING WANG ◽  
HOUYE LIU ◽  
YONGJI TAN
Keyword(s):  
Epidemic Model ◽  
Control Parameter ◽  
Pattern Selection ◽  
Turing Pattern ◽  
Amplitude Equations ◽  
General Survey ◽  
Cross Diffusion ◽  
Model Dynamics ◽  
Flux Boundary Conditions ◽  
Spatial Epidemic Model

In this paper, we have presented Turing pattern selection in a spatial epidemic model with zero-flux boundary conditions, for which we have given a general survey of Hopf and Turing bifurcations, and have derived amplitude equations for the excited modes. Furthermore, we present novel numerical evidence of typical Turing patterns, and find that the model dynamics exhibits complex pattern replication: on increasing the control parameter r, the sequence "H0-hexagons → H0-hexagon-stripe mixtures → stripes → Hπ-hexagon-stripe mixtures → Hπ-hexagons" is observed. This may enrich the research of the pattern formation in diffusive epidemic models.

Turing pattern selection in a reaction-diffusion epidemic model

2011 ◽  
Vol 20 (7) ◽  
pp. 074702 ◽  
Author(s):  
Wei-Ming Wang ◽  
Hou-Ye Liu ◽  
Yong-Li Cai ◽  
Zhen-Qing Li
Keyword(s):  
Epidemic Model ◽  
Reaction Diffusion ◽  
Pattern Selection ◽  
Turing Pattern

scholarly journals Pattern Formation in a Cross-Diffusive Holling-Tanner Model

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Weiming Wang ◽  
Zhengguang Guo ◽  
R. K. Upadhyay ◽  
Yezhi Lin
Keyword(s):  
Pattern Formation ◽  
Sufficient Conditions ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Diffusion Controlled ◽  
Model Dynamics ◽  
Flux Boundary Conditions ◽  
Spatially Distributed

We present a theoretical analysis of the processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; Hopf and Turing bifurcation in a spatial domain is presented, too. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replication. And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive model.

Burgers turbulence model for a channel flow

1971 ◽  
Vol 47 (2) ◽  
pp. 321-335 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Channel Flow ◽  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Amplitude Equations ◽  
Fourier Amplitude ◽  
Navier Stokes Equations ◽  
Burgers Model ◽  
Model Dynamics

The truncated Burgers models have a unique equilibrium state which is defined continuously for all the Reynolds numbers and attainable from a realizable class of initial disturbances. Hence, they represent a sequence of convergent approximations to the original (untruncated) Burgers problem. We have pointed out that consideration of certain degenerate equilibrium states can lead to the successive turbulence-turbulence transitions and finite-jump transitions that were suggested by Case & Chiu. As a prototype of the Navier–Stokes equations, Burgers model can simulate the initial-value type of numerical integration of the Fourier amplitude equations for a turbulent channel flow. Thus, the Burgers model dynamics display certain idiosyncrasies of the actual channel flow problem described by a truncated set of Fourier amplitude equations, which includes only a modest number of modes due to the limited capability of the computer at hand.

scholarly journals Parameter Scaling for Epidemic Size in a Spatial Epidemic Model with Mobile Individuals

PLoS ONE ◽  
2016 ◽  
Vol 11 (12) ◽  
pp. e0168127 ◽  
Author(s):  
Chiyori T. Urabe ◽  
Gouhei Tanaka ◽  
Kazuyuki Aihara ◽  
Masayasu Mimura
Keyword(s):  
Epidemic Model ◽  
Epidemic Size ◽  
Spatial Epidemic Model ◽  
Spatial Epidemic

Limit theorems for the total size of a spatial epidemic

1997 ◽  
Vol 34 (3) ◽  
pp. 698-710 ◽  
Author(s):  
Håkan Andersson ◽  
Boualem Djehiche
Keyword(s):  
Asymptotic Behaviour ◽  
Epidemic Model ◽  
Limit Theorems ◽  
Total Size ◽  
Spatial Epidemic Model ◽  
Spatial Epidemic ◽  
Number Of Individuals ◽  
General Stochastic

We study the long-term behaviour of a sequence of multitype general stochastic epidemics, converging in probability to a deterministic spatial epidemic model, proposed by D. G. Kendall. More precisely, we use branching and deterministic approximations in order to study the asymptotic behaviour of the total size of the epidemics as the number of types and the number of individuals of each type both grow to infinity.

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